Oscillation criteria for elliptic equations
V.B Headley, Charles A. Swanson · Pacific Journal of Mathematics · 1968
Conditions on the coefficients of a linear elliptic partial differential equation will be obtained which are sufficient for the equation to be oscillatory in certain unbounded domains. The criteria obtained in the first three theorems involve in-tegrals of suitable majorants of the coefficients while the criterion in Theorem 4 involves limits of these majorants at infinity. We also obtain a nonoscillation criterion involving similar limits. Oscillation criteria of both limit type and integral type will be obtained for the linear elliptic partial differential equation ( 1) k = Σ DίicLijDjU) + bu = 0 in unbounded domains R in-^-dimensional Euclidean space En. Our theorems constitute extensions of several well-known one-dimensional oscillation theorems of Kneser-Hille [6] (limit type), Leigh ton [8], Moore [10], and Wintner [13] (integral type). A special case of Theorem 4 below was obtained by Glazman [4, 5] when L is the Schrbdinger operator and R coincides with En. Analogues of Theorem 1 were obtained by Kreith [7] and Swanson [12] in the case that one variable is separable and R is limit cylindrical, i.e., contains an in-finitely long cylinder. Points in En are denoted by x = (a;1, x2, , xn) and differentiation with respect to x * is denoted by Di9 i = 1, 2, , n. The functions a{j and b involved in (1) are assumed to be real-valued and continuous on R (J dR, and the matrix {aiά) is supposed to be symmetric and positive definite in R (ellipticity condition). A "solution " of (1) is defined in the usual way [1, 12]. We assume that R contains the origin and that R is large enough at co in the xn direction to contain the cone C a = {x e En: xn ^ | x | cos a} for some a, 0 < a ^ π. The boundary dR of R is supposed to have a piece wise continuous unit normal vector at each point. The follow-ing notations will be used: